28,369 research outputs found
Minimal Triangulations of Manifolds
In this survey article, we are interested on minimal triangulations of closed
pl manifolds. We present a brief survey on the works done in last 25 years on
the following: (i) Finding the minimal number of vertices required to
triangulate a given pl manifold. (ii) Given positive integers and ,
construction of -vertex triangulations of different -dimensional pl
manifolds. (iii) Classifications of all the triangulations of a given pl
manifold with same number of vertices.
In Section 1, we have given all the definitions which are required for the
remaining part of this article. In Section 2, we have presented a very brief
history of triangulations of manifolds. In Section 3, we have presented
examples of several vertex-minimal triangulations. In Section 4, we have
presented some interesting results on triangulations of manifolds. In
particular, we have stated the Lower Bound Theorem and the Upper Bound Theorem.
In Section 5, we have stated several results on minimal triangulations without
proofs. Proofs are available in the references mentioned there.Comment: Survey article, 29 page
Transforming triangulations on non planar-surfaces
We consider whether any two triangulations of a polygon or a point set on a
non-planar surface with a given metric can be transformed into each other by a
sequence of edge flips. The answer is negative in general with some remarkable
exceptions, such as polygons on the cylinder, and on the flat torus, and
certain configurations of points on the cylinder.Comment: 19 pages, 17 figures. This version has been accepted in the SIAM
Journal on Discrete Mathematics. Keywords: Graph of triangulations,
triangulations on surfaces, triangulations of polygons, edge fli
Simulating Four-Dimensional Simplicial Gravity using Degenerate Triangulations
We extend a model of four-dimensional simplicial quantum gravity to include
degenerate triangulations in addition to combinatorial triangulations
traditionally used. Relaxing the constraint that every 4-simplex is uniquely
defined by a set of five distinct vertexes, we allow triangulations containing
multiply connected simplexes and distinct simplexes defined by the same set of
vertexes. We demonstrate numerically that including degenerated triangulations
substantially reduces the finite-size effects in the model. In particular, we
provide a strong numerical evidence for an exponential bound on the entropic
growth of the ensemble of degenerate triangulations, and show that a
discontinuous crumpling transition is already observed on triangulations of
volume N_4 ~= 4000.Comment: Latex, 8 pages, 4 eps-figure
Regular triangulations of dynamic sets of points
The Delaunay triangulations of a set of points are a class of
triangulations which play an important role in a variety of
different disciplines of science. Regular triangulations are a
generalization of Delaunay triangulations that maintain both their
relationship with convex hulls and with Voronoi diagrams. In regular
triangulations, a real value, its weight, is assigned to each point.
In this paper a simple data structure is presented that allows
regular triangulations of sets of points to be dynamically updated,
that is, new points can be incrementally inserted in the set and old
points can be deleted from it. The algorithms we propose for
insertion and deletion are based on a geometrical interpretation of
the history data structure in one more dimension and use lifted
flips as the unique topological operation. This results in rather
simple and efficient algorithms. The algorithms have been
implemented and experimental results are given.Postprint (published version
Counting Triangulations and other Crossing-Free Structures Approximately
We consider the problem of counting straight-edge triangulations of a given
set of points in the plane. Until very recently it was not known
whether the exact number of triangulations of can be computed
asymptotically faster than by enumerating all triangulations. We now know that
the number of triangulations of can be computed in time,
which is less than the lower bound of on the number of
triangulations of any point set. In this paper we address the question of
whether one can approximately count triangulations in sub-exponential time. We
present an algorithm with sub-exponential running time and sub-exponential
approximation ratio, that is, denoting by the output of our
algorithm, and by the exact number of triangulations of , for some
positive constant , we prove that . This is the first algorithm that in sub-exponential time computes a
-approximation of the base of the number of triangulations, more
precisely, . Our algorithm can be
adapted to approximately count other crossing-free structures on , keeping
the quality of approximation and running time intact. In this paper we show how
to do this for matchings and spanning trees.Comment: 19 pages, 2 figures. A preliminary version appeared at CCCG 201
Generating families of surface triangulations. The case of punctured surfaces with inner degree at least 4
We present two versions of a method for generating all triangulations of any
punctured surface in each of these two families: (1) triangulations with inner
vertices of degree at least 4 and boundary vertices of degree at least 3 and
(2) triangulations with all vertices of degree at least 4. The method is based
on a series of reversible operations, termed reductions, which lead to a
minimal set of triangulations in each family. Throughout the process the
triangulations remain within the corresponding family. Moreover, for the family
(1) these operations reduce to the well-known edge contractions and removals of
octahedra. The main results are proved by an exhaustive analysis of all
possible local configurations which admit a reduction.Comment: This work has been partially supported by PAI FQM-164; PAI FQM-189;
MTM 2010-2044
Planar maps, circle patterns and 2d gravity
Via circle pattern techniques, random planar triangulations (with angle
variables) are mapped onto Delaunay triangulations in the complex plane. The
uniform measure on triangulations is mapped onto a conformally invariant
spatial point process. We show that this measure can be expressed as: (1) a sum
over 3-spanning-trees partitions of the edges of the Delaunay triangulations;
(2) the volume form of a K\"ahler metric over the space of Delaunay
triangulations, whose prepotential has a simple formulation in term of ideal
tessellations of the 3d hyperbolic space; (3) a discretized version (involving
finite difference complex derivative operators) of Polyakov's conformal
Fadeev-Popov determinant in 2d gravity; (4) a combination of Chern classes,
thus also establishing a link with topological 2d gravity.Comment: Misprints corrected and a couple of footnotes added. 42 pages, 17
figure
Degree-regular triangulations of torus and Klein bottle
A triangulation of a connected closed surface is called weakly regular if the
action of its automorphism group on its vertices is transitive. A triangulation
of a connected closed surface is called degree-regular if each of its vertices
have the same degree. Clearly, a weakly regular triangulation is
degree-regular. In 1999, Lutz has classified all the weakly regular
triangulations on at most 15 vertices. In 2001, Datta and Nilakantan have
classified all the degree-regular triangulations of closed surfaces on at most
11 vertices.
In this article, we have proved that any degree-regular triangulation of the
torus is weakly regular. We have shown that there exists an -vertex
degree-regular triangulation of the Klein bottle if and only if is a
composite number . We have constructed two distinct -vertex weakly
regular triangulations of the torus for each and a -vertex weakly regular triangulation of the Klein bottle for each . For , we have classified all the -vertex
degree-regular triangulations of the torus and the Klein bottle. There are
exactly 19 such triangulations, 12 of which are triangulations of the torus and
remaining 7 are triangulations of the Klein bottle. Among the last 7, only one
is weakly regular.Comment: Revised version, 26 pages, To appear in Proceedings of Indian Academy
of Sciences (Math. Sci.
Entropy of unimodular Lattice Triangulations
Triangulations are important objects of study in combinatorics, finite
element simulations and quantum gravity, where its entropy is crucial for many
physical properties. Due to their inherent complex topological structure even
the number of possible triangulations is unknown for large systems. We present
a novel algorithm for an approximate enumeration which is based on calculations
of the density of states using the Wang-Landau flat histogram sampling. For
triangulations on two-dimensional integer lattices we achive excellent
agreement with known exact numbers of small triangulations as well as an
improvement of analytical calculated asymptotics. The entropy density is
consistent with rigorous upper and lower bounds. The presented
numerical scheme can easily be applied to other counting and optimization
problems.Comment: 6 pages, 7 figure
Non-geometric veering triangulations
Recently, Ian Agol introduced a class of "veering" ideal triangulations for
mapping tori of pseudo-Anosov homeomorphisms of surfaces punctured along the
singular points. These triangulations have very special combinatorial
properties, and Agol asked if these are "geometric", i.e. realised in the
complete hyperbolic metric with all tetrahedra positively oriented. This paper
describes a computer program Veering, building on the program Trains by Toby
Hall, for generating these triangulations starting from a description of the
homeomorphism as a product of Dehn twists. Using this we obtain the first
examples of non-geometric veering triangulations; the smallest example we have
found is a triangulation with 13 tetrahedra
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